Volume of Cone

It was easy to determine the expressions for the volumes of a cuboid and a cylinder with much rigorous analysis. However, the expression for the volume of a cone is not that straightforward to obtain. This is because the cross-sectional area of the cone is changing as we move along its height. Hence, we cannot use the relation Base Area × Height to obtain a cone’s volume.

We will use a more involved technique to evaluate the volume of a cone. The discussion which follows might be slightly challenging, so you are urge to read it multiple times until you have complete clarity.

As we said earlier, the problem with using Base Area × Height for a cone is that the cross-sectional area of the cone changes as we move along its height. However, suppose that we divide the cone into a large number of slices. Now, observe any one slice:

Can we calculate the volume of this slice? Note that even this slice is not a right circular cylinder, since the curved edges of the slice are not perpendicular to the base. However, since the thickness of the slice is very small, we can approximate the slice as a cylinder:

If the radius of the lower base of the slice is r and its thickness is h, we can approximate the slice as a right circular cylinder of base radius r and height h. This is possible because the difference between the volumes of the slice and the cylinder will be small, as the figure above shows (the smaller the thickness of the slice, the smaller will this difference be).

Having understood this, think about how we can calculate the volume of the cone (take its radius to be R and height to be H). Simple! We cut up the cone into a (large) number of such slices, measure the volume of each slice, and then add up all these volumes. The larger the number of slices, the more accurate our answer will be (because for each slice, our calculated volume will be closer to the actual volume).

Suppose that we divide the cone into N slices, where N is some large number. Number the slices from 1 to N, starting from the lowermost slice (the topmost slice is actually a cone):

Focus your attention on the ith slice. What is the radius of (the base of) this slice? What is its thickness? To determine the thickness is simple. The height of the cone is H, and we have divided it into N slices of equal thickness. Thus, the thickness of each slide is \(d = \frac{H}{N}\). To calculate its radius r, note that the ith slice has \(\left( {i - 1} \right)\) slices below it, with each slice being of thickness d. Thus, the height of the ith slice above the base is \(\left( {i - 1} \right)d\), as the following figure shows:

Now, for the final step, which also happens to be really important! Remember how we talked about approximating the volume of any slice using the volume of a cylinder of the same dimensions. The approximation will get better and better as the thickness of the slice reduces.

This means that if we cut up the cone into a really large number of slices (if N is really large), and add up the volumes of these slices, we will obtain a value very close to the actual value of the cone.

If Ngoes to infinity, we will obtain the exact volume of the cone. However, if \(N \to \infty \), then \(\frac{1}{N} \to 0\) (this simple means that if N is really large, then \(\frac{1}{N}\) is really small – close to 0). Thus, the exact volume of the cone can be calculated by substituting \(\frac{1}{N} \to 0\) in the expression above: